The proof follows from the dual representations of \(H_1^*\)  and \(H_2^*\)  in Lemma REF  and a measurable selection argument as in [1]} Theorem 12.1.10. For every competitor \((m,\frac{d\omega }{dm},\frac{dW}{dm})\)  such that \(H_1^*(x, \frac{d\omega }{dt\otimes dm}(t,x)) +H_2^*(x, \frac{dW}{dt\otimes dm}(t,x)) < +\infty \)  one can find measurable functions \(\mathbf {q}_{A_1} : [0,T]\times \mathbb {R}^d \rightarrow \mathcal {P}_1(A_1)\)  and \(\mathbf {q}_{A_2} : [0,T]\times \mathbb {R}^d \rightarrow \mathcal {P}(A_2)\)  such that, for every \((t,x) \in [0,T]\times \mathbb {R}^d\)  one has:
\( H_1^*(x,\frac{d\omega }{dt \otimes dm}(t,x)) = \int _{A_1} f_1(x,a^1) \mathbf {q}_{A_1}(t,x)(da^1) \mbox{, } H_2^*(x,\frac{dW}{dt \otimes dm}(t,x)) = \int _{A_2} f_2(x,a^2) \mathbf {q}_{A_2}(t,x)(da^2) \) 
